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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Generalized Moser lemma


Author: Mathieu Stiénon
Journal: Trans. Amer. Math. Soc. 362 (2010), 5107-5123
MSC (2010): Primary 53C15, 17B62, 17B66
Published electronically: May 10, 2010
MathSciNet review: 2657674
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Abstract: We show how the classical Moser lemma from symplectic geometry extends to generalized complex structures (GCS) on arbitrary Courant algebroids. For this, we extend the notion of a Lie derivative to sections of the tensor bundle $ (\otimes^i E)\otimes(\otimes^j E^*)$ with respect to sections of the Courant algebroid $ E$ using the Dorfman bracket. We then give a cohomological interpretation of the existence of one-parameter families of GCS on $ E$ and of flows of automorphims of $ E$ identifying all GCS of such a family. In the particular case of symplectic manifolds, we recover the results of Moser. Finally, we give a criterion to detect the local triviality of arbitrary GCS which generalizes the Darboux-Weinstein theorem.


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Additional Information

Mathieu Stiénon
Affiliation: Departement Mathematik, E.T.H. Zürich, 8092 Zürich, Switzerland
Address at time of publication: Institut de Mathématiques de Jussieu, Université Paris-Diderot, 75013 Paris, France
Email: stienon@math.ethz.ch, stienon@math.jussieu.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04965-2
PII: S 0002-9947(10)04965-2
Keywords: Courant algebroid, Moser lemma, generalized complex structure, Dorfman bracket, symplectic form
Received by editor(s): April 4, 2008
Published electronically: May 10, 2010
Additional Notes: This work was supported by the European Union through the FP6 Marie Curie RTN ENIGMA (Contract number MRTN-CT-2004-5652) and by the E.S.I. Vienna through a Junior Research Fellowship.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.